1. Field of the Invention
The present invention relates to digital signal processing techniques and signal filtering, and particularly to a system and method for least mean fourth adaptive filtering that provides an adaptive filter and method of adaptive filtering utilizing a normalized least mean fourth algorithm.
2. Description of the Related Art
The least mean fourth (LMF) algorithm has uses in a wide variety of applications. The algorithm outperforms the well-known least mean square (LMS) algorithm in cases with non-Gaussian noise. Even in Gaussian environments, the LMF algorithm can outperform the LMS algorithm when initialized far from the Wiener solution. The true usefulness of the LMF algorithm lies in its faster initial convergence and lower steady-state error relative to the LMS algorithm. More importantly, its mean-fourth error cost function yields better performance than that of the LMS for noise of sub-Gaussian nature, or noise with a light-tailed probability density function. However, this higher-order algorithm requires a much smaller step-size to ensure stable adaptation, since the cubed error in the LMF gradient vector can cause devastating initial instability, resulting in unnecessary performance degradation.
One approach to solving the above degradation problem is to normalize the weight update term of the algorithm. In conventional techniques, the LMF algorithm is normalized by dividing the weight vector update term by the squared norm of the regressor. In some prior art techniques, this normalization is performed by dividing the weight vector update term by a weighted sum of the squared norm of the regressor and the squared norm of the estimation error vector. Thus, the LMF algorithm is normalized by both the signal power and the error power. Combining the signal power and the error power has the advantage that the former normalizes the input signal, while the latter can dampen down the outlier estimation errors, thus improving stability while still retaining fast convergence.
The above has been modified by using an adaptive, rather than fixed, mixing parameter in the weighted sum of the squared norm of the regressor and the squared norm of the estimation error vector. The adaptation of the mixing parameter improves the tracking properties of the algorithm. However, unlike the normalization of the LMS algorithm, the above normalization techniques of the LMF algorithm do not protect the algorithm from divergence when the input power of the adaptive filter increases. In fact, as will be shown below, the above prior art normalized LMF (NLMF) algorithms diverge when the input power of the adaptive filter exceeds a threshold that depends on the step-size of the algorithm. The reason for this drawback is that in all of the above techniques, the weight vector update term, which is a fourth order polynomial of the regressor, is normalized by a second order polynomial of the regressor.
The prior art normalization techniques do not ensure a normalization of the input signal. Thus, the algorithm stability remains dependent on the input power of the adaptive filter.
For the particular application of adaptive plant identification, as diagrammatically illustrated in FIG. 1, the output ak of the unknown plant 100 is given by:ak=gTxk+bk  (1)whereg≡(g1,g2, . . . ,gN)T  (2)is a vector composed of the plant parameters gi (from an unknown finite impulse response (FIR) filter 102), andxk=(xk,xk−1,xk−2, . . . ,xk−N+1)T  (3)is the regressor vector at time k. N is the number of plant parameters, xk is the plant input, bk is the plant noise, and the notation (.)T represents the transpose of (.). The identification of the plant is made by an adaptive finite impulse response (FIR) filter 104 whose length is assumed equal to that of the plant. The weight vector hk of the adaptive filter is adapted on the basis of the error ek, which is given byek=ak−hkTxk,  (4)where hk=(h1,k, h2,k, . . . , hN,k)T. The adaptation algorithm of interest is the LMF algorithm, which is described byhk+1=hk+ƒek3xk,  (5)where μ>0 is the algorithm step-size. The error signal ek can be decomposed into two terms as follows:ek≡bk+εk.  (6)
The first term on the right hand side of equation (6), bk, is the plant noise. The second term, εk, is the excess estimation error. The weight deviation vector is defined byvk≡hk−g.  (7)From equations (1), (4), (6) and (7),εk=−vkTxk.  (8)
Inserting equations (1), (4) and (7) into equation (5) yieldsvk+1=vk+μ(bk−vkTxk)3xk.  (9)
In the above, the following assumptions are used: The first assumption (assumption A1) is that the sequences {xk} and {bk} are mutually independent. The second assumption (assumption A2) is that {xk} is a stationary sequence of zero mean random variables with a finite variance σx2. The third assumption (assumption A3) is that {bk} is a stationary sequence of independent zero mean random variables with a finite variance σb2. Such assumptions are typical in the context of adaptive filtering.
In order to emphasize the need for normalization in the least mean fourth algorithm, examining the normalization of the LMS algorithm is important. The stability of the LMS algorithm is dependent upon the input power of the adaptive filter. This makes it very hard, if not impossible, to choose a step-size that guarantees stability of the algorithm when there is lack of knowledge about the input power. This is solved by normalizing the weight update term by ∥xk∥2, where ∥xk∥ is the Euclidean norm of the vector xk, which is defined as ∥xk∥=√{square root over (xkTxk)}. The resulting algorithm is referred to as the normalized LMS (NLMS) algorithm. This algorithm is stable for all values of the filter input power so long as the step-size is between 0 and 2.
It is desirable to develop a version of the LMF algorithm that has a similar feature as that of the NLMS algorithm; i.e., stability for all values of the filter input power for an appropriate fixed range of the step-size. One prior art normalization technique is given as
                                          v                          k              +              1                                =                                    v              k                        +                                                            μ                  ⁡                                      (                                                                  b                        k                                            -                                                                        v                          k                          T                                                ⁢                                                  x                          k                                                                                      )                                                  3                            ⁢                                                x                  k                                                                                                                x                      k                                                                            2                                                                    ,                            (        10        )            and a second version is given by
                                          v                          k              +              1                                =                                    v              k                        +                                                            μ                  ⁡                                      (                                                                  b                        k                                            -                                                                        v                          k                          T                                                ⁢                                                  x                          k                                                                                      )                                                  3                            ⁢                                                x                  k                                                  δ                  +                                      λ                    ⁢                                                                                                                    x                          k                                                                                            2                                                        +                                                            (                                              1                        -                        λ                                            )                                        ⁢                                                                                                                    e                          k                                                                                            2                                                                                                          ,                            (        11        )            where δ is a small positive number, 0<δ<1, and ek=(ek, ek−1, ek−2, . . . , ek−N+1)T is the error vector. The parameter λ is referred to as the mixing power parameter. The choice of λ is a compromise between fast convergence and low steady-state error. However, the stability of the above algorithms depends on the mean square input of the adaptive filter. To show this undesired feature for the algorithm of equation (10), we may consider the scalar case, N=1, with zero noise, bk=0, and binary input, xk ε{−1,1}, with μ=0.5. In this case, equation (10) implies that:vk+1=vk−0.5vk3.  (12)
If v1=1, then equation (12) implies that v2=0.5, v3=0.4375, v4=0.3956, v5=0.3647, etc. Thus, vk is decaying in this case. Repeating this example with xk ε{−4,4}, while keeping all other conditions unchanged, equation (10) implies that:vk+1=vk−8vk3.  (13)
Again, if v1=1, then equation (13) yields v2=−7, v3=2737, v4=1.6(1011), v5=3.5(1034), etc. Thus, the algorithm of equation (10) diverges in this case. This shows that the stability of the normalized LMF algorithm of equation (10) depends on the input power of the adaptive filter.
It can also be shown that the stability of the algorithm of equation (11) also depends on the input power. We may consider again the scalar case with μ=0.5, δ=0, λ=0.5, xk ε{−1,1}, and bk=0. In this case, equations (6) and (8) imply that ek2=vk2xk2 and equation (11) implies that:
                              v                      k            +            1                          =                              v            k                    -                                                    v                k                3                                            1                +                                  v                  k                  2                                                      .                                              (        14        )            
If v1=1, then equation (14) produces v2=0.5, v3=0.4, v4=0.3448, v5=0.3082, etc. Thus, vk is decaying in this case. Repeating this example with xk ε{−4,4}, while keeping all other conditions unchanged, equation (11) implies that:
                              v                      k            +            1                          =                              v            k                    -                                                    16                ⁢                                                                  ⁢                                  v                  k                  3                                                            1                +                                  v                  k                  2                                                      .                                              (        15        )            
Again, if v1=1, then equation (15) produces v2=−7, v3=102.76, v4=1541.2, v5=23119, etc. Thus, the algorithm of equation (11) diverges in this case. This shows that the stability of the normalized LMF algorithm of equation (11) depends on the input power of the adaptive filter.
The above results regarding the dependence of the stability of the prior art NLMF algorithms on the input power of the adaptive filter suggest the need for an NLMF algorithm whose stability does not depend on the input power. Thus, a system and method for least mean fourth adaptive filtering solving the aforementioned problems is desired.